Meaning of a union symbol in front of a set?

In summary, the conversation discusses the use of the union operator in a function definition on page 75 of the book "Automata, Computability, and Complexity" by Elaine Rich. The person asking the question is unsure of its meaning and asks for clarification. The expert explains that the union symbol is used when the elements in the set are sets themselves and provides an example to illustrate this concept. The conversation ends with the person thanking the expert for their explanation.
  • #1
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I'm trying to read this book "Automata, Computability, and Complexity" by Elaine Rich and on page 75 it defines this function: [itex]\delta'(Q,c) = \cup\{eps(p):\exists q\in Q((q,c,p)\in\Delta)\}[/itex]
I've never seen the union operator used in this way. What does it mean?
Apologies if this is in the wrong section.

Edit: I don't care what the stuff inside the brackets means. I understand that part. I'm asking, what does [itex]\cup[/itex] mean when it's front of any set? It could be something like [itex]\cup\{x:x\in\mathbb{R}\}[/itex]
 
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  • #2
A union in front of a set is written when the elements in the set are sets themselves. So for example, we can have

[tex]\bigcup \{A~\vert~A\in \mathcal{A}\}[/tex]

this just means to take the union of each element in the set. That is:

[tex]\bigcup_{A\in \mathcal{A}}{A}[/tex]

Writing [itex]\cup \{x~\vert~x\in \mathbb{R}\}[/itex] doesn't make much sense since the element of [itex]\mathbb{R}[/itex] aren't sets (unless you see them as Dedekind cuts).
 
  • #3
That explains things. Thanks!
 

1. What does the union symbol in front of a set mean?

The union symbol (∪) in front of a set indicates that the set is being combined with another set. It represents the combination of all elements from both sets, without any duplicates.

2. How is the union of two sets represented mathematically?

The union of two sets A and B is represented as A ∪ B. This means that all elements in A and B will be included in the new set without any duplicates.

3. What is the difference between union and intersection of sets?

The union (∪) of two sets combines all elements from both sets, while the intersection (∩) of two sets only includes elements that are common to both sets.

4. Can the union of two sets contain duplicate elements?

No, the union of two sets cannot contain duplicate elements. If an element is present in both sets, it will only be included once in the union set.

5. How is the union of multiple sets calculated?

The union of multiple sets can be calculated by combining all the elements from each set into one set, without any duplicates. This can be represented as A ∪ B ∪ C... where A, B, and C are individual sets.

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